Pade Approximation (further generalizations?---feature request)
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From: telefunkenvf14 on 14 Apr 2010 05:16 I've been playing around with PadeApproximant[] in Mathematica and have been really impressed at the accuracy of the approach. According to Wikipedia: A Pad=E9 approximant approximates a function in one variable. An approximant in two variables is called a Chisholm approximant, in multiple variables a Canterbury approximant (after Graves-Morris at the University of Kent). Does anyone know if v8 will include Chisholm and Canterbury approximation? -RG
From: David Reiss on 14 Apr 2010 23:39 Another addition to the Pade Approximant function that would be very useful would be the generalized Pade Approximant: these are Pade Approximants that are based on more than one expansion point. I coded this up many years ago for some work in Radar propagation analysis (never published but I really should have...): http://scientificarts.com/radar/radar/PadeMethod/index.html more stuff on Radar is here... which I really should do something commercial with sometime.... http://scientificarts.com/radar/radar/index.html --David http://scientificarts.com/worklife On Apr 14, 5:16 am, telefunkenvf14 <rgo...(a)gmail.com> wrote: > I've been playing around with PadeApproximant[] in Mathematica and have b= een > really impressed at the accuracy of the approach. > > According to Wikipedia: > > A Pad=E9 approximant approximates a function in one variable. An > approximant in two variables is called a Chisholm approximant, in > multiple variables a Canterbury approximant (after Graves-Morris at > the University of Kent). > > Does anyone know if v8 will include Chisholm and Canterbury > approximation? > > -RG
From: telefunkenvf14 on 16 Apr 2010 05:49 On Apr 14, 10:39 pm, David Reiss <dbre...(a)gmail.com> wrote: > Another addition to the Pade Approximant function that would be very > useful would be the generalized Pade Approximant: these are Pade > Approximants that are based on more than one expansion point. I coded > this up many years ago for some work in Radar propagation analysis > (never published but I really should have...): > > http://scientificarts.com/radar/radar/PadeMethod/index.html > > more stuff on Radar is here... which I really should do something > commercial with sometime.... > > http://scientificarts.com/radar/radar/index.html > > --Davidhttp://scientificarts.com/worklife > > On Apr 14, 5:16 am, telefunkenvf14 <rgo...(a)gmail.com> wrote: > > > > > I've been playing around with PadeApproximant[] in Mathematica and have= b= > een > > really impressed at the accuracy of the approach. > > > According to Wikipedia: > > > A Pad=E9 approximant approximates a function in one variable. An > > approximant in two variables is called a Chisholm approximant, in > > multiple variables a Canterbury approximant (after Graves-Morris at > > the University of Kent). > > > Does anyone know if v8 will include Chisholm and Canterbury > > approximation? > > > -RG Good timing! I actually came across a economics paper last night that used Pade at more than one point, so it would be nice to see how this could be coded in Mathematica. Could you show me? (I understand if you don't want to, or if there are too many other dependencies on other package functions.) -RG
From: David Reiss on 17 Apr 2010 06:05 On Apr 16, 5:49 am, telefunkenvf14 <rgo...(a)gmail.com> wrote: > On Apr 14, 10:39 pm, David Reiss <dbre...(a)gmail.com> wrote: > > > > > > > Another addition to the Pade Approximant function that would be very > > useful would be the generalized Pade Approximant: these are Pade > > Approximants that are based on more than one expansion point. I coded > > this up many years ago for some work in Radar propagation analysis > > (never published but I really should have...): > > >http://scientificarts.com/radar/radar/PadeMethod/index.html > > > more stuff on Radar is here... which I really should do something > > commercial with sometime.... > > >http://scientificarts.com/radar/radar/index.html > > > --Davidhttp://scientificarts.com/worklife > > > On Apr 14, 5:16 am, telefunkenvf14 <rgo...(a)gmail.com> wrote: > > > > I've been playing around with PadeApproximant[] in Mathematica and have > been really impressed at the accuracy of the approach. > > > > According to Wikipedia: > > > > A Pad=E9 approximant approximates a function in one variable. An > > > approximant in two variables is called a Chisholm approximant, in > > > multiple variables a Canterbury approximant (after Graves-Morris at > > > the University of Kent). > > > > Does anyone know if v8 will include Chisholm and Canterbury > > > approximation? > > > > -RG > > Good timing! I actually came across a economics paper last night that > used Pade at more than one point, so it would be nice to see how this > could be coded in Mathematica. > > Could you show me? (I understand if you don't want to, or if there are > too many other dependencies on other package functions.) > > -RG Let me see if I can track it down and if it is actually usable out of its original context ... it may not be pretty! I've learned a lot in the last 10 years! A good place to read up on it though is in Bender and Orszag if my memory serves me right: http://www.amazon.com/Advanced-Mathematical-Methods-Scientists-Engineers/dp/0387989315 --David
From: David Reiss on 18 Apr 2010 05:58
On Apr 17, 6:05 am, David Reiss <dbre...(a)gmail.com> wrote: > On Apr 16, 5:49 am, telefunkenvf14 <rgo...(a)gmail.com> wrote: > > > > > > > On Apr 14, 10:39 pm, David Reiss <dbre...(a)gmail.com> wrote: > > > > Another addition to the Pade Approximant function that would be very > > > useful would be the generalized Pade Approximant: these are Pade > > > Approximants that are based on more than one expansion point. I coded > > > this up many years ago for some work in Radar propagation analysis > > > (never published but I really should have...): > > > >http://scientificarts.com/radar/radar/PadeMethod/index.html > > > > more stuff on Radar is here... which I really should do something > > > commercial with sometime.... > > > >http://scientificarts.com/radar/radar/index.html > > > > --Davidhttp://scientificarts.com/worklife > > > > On Apr 14, 5:16 am, telefunkenvf14 <rgo...(a)gmail.com> wrote: > > > > > I've been playing around with PadeApproximant[] in Mathematica and = have > > been really impressed at the accuracy of the approach. > > > > > According to Wikipedia: > > > > > A Pad=E9 approximant approximates a function in one variable. An > > > > approximant in two variables is called a Chisholm approximant, in > > > > multiple variables a Canterbury approximant (after Graves-Morris at > > > > the University of Kent). > > > > > Does anyone know if v8 will include Chisholm and Canterbury > > > > approximation? > > > > > -RG > > > Good timing! I actually came across a economics paper last night that > > used Pade at more than one point, so it would be nice to see how this > > could be coded in Mathematica. > > > Could you show me? (I understand if you don't want to, or if there are > > too many other dependencies on other package functions.) > > > -RG > > Let me see if I can track it down and if it is actually usable out of > its original context ... it may not be pretty! I've learned a lot i= n > the last 10 years! A good place to read up on it though is in Bender > and Orszag if my memory serves me right: > > http://www.amazon.com/Advanced-Mathematical-Methods-Scientists-Engine... > > --David Alas, it was a rather rambling notebook with a special case computation for the expansion of the solution of a particular differential equation around several points. So the code is not usable by anyone else -- and perhaps not me either anymore! Notebook archeology.... |